The problem asks by Green's Theorem as opposed to the standard method of calculating the the line integral.

You need to find,
Another notation for this (if that confuses you) is,

So the vector field is,
Let, and
By Green's theorem,
Where,
And,
So the double integral over that closed region is,

Can you take it from there?
(Hint: Divide your integral into two regions one is the square and the other is a semicircle.)
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(For Engineering Students)
There happens to be a cool way to get that value if you ever taken an engineering course. You should recognize that is the first-moment. And hence you need to find the centroid of this shape. Which is simple for it is one of the standard shapes used.

Ok, so why does he (the professor) have us parametrizing each of the paths and then intergrating them individually and adding them together.

I am asking because what you have shown makes sense and its simpler.

As always I thank you for taking time to answer my questions it is really appreciated.

I think your professor wanted to do this problem in two ways. One, the standard way of parametrizing the simply connected positevly oriented piecewise smooth closed curve (this is one of my favorite expressions in math, just look how long it is and how cool it sounds). Second, by using Green's theorem because this is a simply connected positevly oriented piecewise smooth closed curve. And show that you get the same result in the end.